Modal interactions of a geometrically nonlinear sandwich beam with transversely compressible core

Thin-Walled Structures 73 (2013) 242–251

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Modal interactions of a geometrically nonlinear sandwich beam with transversely compressible core Zhanming Qin n, Yizhe Feng, Guiping Zhao State Key Lab for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, No. 28 West Xianning Rd, Xi'an 710049, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 1 November 2012 Received in revised form 25 July 2013 Accepted 25 July 2013 Available online 20 September 2013

Modal interactions of a geometrically nonlinear sandwich beam with transversely compressible core in the presence of combination internal resonance are investigated. At first, a geometrically nonlinear, {2,1}-order theory is used to derived the equations of motion and the compatible boundary conditions of the beam. Then, Galerkin's weighted residual method and the multiscale approach are used to address the governing system. Next, modal interactions in the presence of combination internal resonance and subjected to primary-resonance excitation are investigated. Finally, the commercial code ABAQUS is used to validate the theoretical results we have obtained. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Sandwich beam Geometrically nonlinear Compressible core Modal interactions

1. Introduction A sandwich panel is a layered structure consisting of two thin face sheets which are bonded to a thick core layer. Within the principle of sandwich construction, the face sheets carry the tangential and bending loads while the core transmits the transverse normal and shear loads. This special configuration brings us an extremely low weight while retaining high bending stiffness thanks to the adopted lightweight low-density core. In view of its low weight, sandwich panels have extensive application in many fields of engineering such as aerospace, aeronautics, automotive, naval construction and civil engineering (see e.g., [1,2]). When the thick core layer is made of a weak material, sandwich panels would have complicated deformations, buckling behavior, and rich dynamic features [3]. For example, different from the standard laminated structures, the buckling modes of a sandwich panel with weak core can be present in two scales: (1) global or overall buckling, which is similar to Euler's buckling for homogeneous columns, and (2) some local forms of buckling of the face sheets called wrinkling (see e.g., [2,4] and the references thereof). Expectantly, similar phenomena would arise in the vibration responses of the sandwich panels. It is well-known that nonlinear structures may display manifold nonlinear characteristics such as multiple solutions, limit cycles, subharmonic and superharmonic resonances, various modal interactions, bifurcations and chaotic motions (see e.g.,

n

Corresponding author. Tel.: þ 86 29 8266 3850; fax: þ 86 29 8266 9044. E-mail address: [email protected] (Z. Qin).

0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.07.014

[5, pp. xv–xviii]). Among these behaviors, modal interactions pose a particular concern since interactions between global and local modes of a sandwich panel due to its soft and thick cores can cause disastrous results if vibrational energy is transferred from low-amplitude high-frequency modes to lower modes with highamplitude. One necessary condition for the presence of modal interaction is that the linear natural frequencies ωi are commensurate or nearly commensurate, i.e., ∑ni¼ 1 ki ωi  0, with ki being positive or negative integers [5, p. xvii]. However, for sandwich panels, this condition can be easily fulfilled and there may even exist numerous groups of linear modes which simultaneously fulfill this condition. As an example, Table 1 shows the first 15 linear natural frequencies of a flat sandwich panel experimentally investigated by [6]. Among these frequencies, the following combinations simultaneously fulfill the nearly commensurable condition: ω21 ð45:0Þ þ ω41 ð133:0Þ ¼ 178:0  ω42 ð177:0Þ, ω12 ð69:0Þ þ ω42 ð177:0Þ ¼ 246:0 ¼ ω14 ð246:0Þ, ω22 ð92:0Þ þ ω23 ð169:0Þ ¼ 261:0  ω24 ð262:0Þ, 2ω21 ð90:0Þ þ ω31 ð78:0Þ ¼ 168:0  ω23 ð169:0Þ, ω12 ð69:0Þ þ ω32 ð129:0Þ ¼ 198:0  ω33 ð199:0Þ, ω21 ð45:0Þ þ ω13 ð152:0Þ ¼ 197:0  ω33 ð199:0Þ. This motivates us to make an in-depth study of the nonlinear vibration responses of sandwich panels. To keep our effort to be focused, we consider in the present paper a straight two-dimensional (2-D) sandwich panel with transversely compressible core, which can be modeled as a sandwich beam, and investigate its nonlinear dynamic responses in the presence of combination internal resonance. Towards this end, an effective semi-analytical and semi-numerical method is used to investigate the modal interaction phenomena of the sandwich beam in the presence of combination internal resonance.

Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

243

Table 1 Eigenfrequencies ωmn (Hz) of a flat sandwich panel. m

n 1

1

2

3

4

5

23.5 23.0 45.1 45.0 45.0 80.7 78.0 80.0 130.0 133.0 129.0 192.3 188.0 191.0

Note 2

3

4

5

71.0 69.0 71.0 92.1 92.0 91.0 126.8 129.0 126.0 174.9 177.0 174.0

146.5 152.0 146.0 166.7 169.0 165.0 200.1 199.0 195.0

245.3 246.0 244.0 264.5 262.0 263.0

362.5 381.0 360.0

Present study [1] (exp.) [1] (num.) Present study [1] (exp.) [1] (num.) Present study [1] (exp.) [1] (num.) Present study [1] (exp.) [1] (num.) Present study [1] (exp.) [1] (num.)

Specifically, a nonlinear sandwich structures theory originally developed by Hohe and Librescu [7] is adopted to investigate the sandwich beam with transversely compressible core. The theory adopts the standard Kirchhoff hypothesis for the face sheets whereas a {2, 1}-order power series expansion is introduced for modeling the core's displacements. After incorporating the nonlinearity of deformation of the sandwich beam, the equations of motion and the compatible boundary conditions are derived from an extended Hamilton's principle [8]. Then, Galerkin's method and the method of multiple scales are adopted to semi-discretize and solve the related nonlinear problems. Next, modal interactions in the presence of combination internal resonance are analytically investigated and the solvability conditions are numerically integrated via the 4th-order Runge–Kutta method (see e.g., [8, pp. 214–222]). Finally, for the purpose of validating the preceding theoretical results, the commercial finite element code ABAQUS [9] is used to simulate the nonlinear vibration responses of the sandwich beam subjected to primary-resonance excitation. The fast Fourier transform (FFT) is further used for the frequencydomain analysis.

2. Kinematics and constitutive relations We will consider a symmetric sandwich beam with respect to its mid-surface (see Fig. 1). The thickness of each of the face sheets and the core are denoted by tf and tc, respectively, while its length is denoted by ℓ. A Cartesian coordinate system x2 x3 is adopted as shown in Fig. 1. For modeling the sandwich beam, we follow the theoretical framework provided in [7], in which, the displacements ui (i¼2,3) of the three layers of the sandwich beam are individually expanded into power series with respect to x3. For the face sheets, the classical Kirchhoff hypothesis is adopted whereas the secondorder power series expansion is used for the core layer's horizontal displacement, and the first-order power series expansion is used for the core layer's vertical displacement. It is termed as the {2,1}order theory according to the name convention proposed in [10]. As a result, the following expressions for the displacement field of the face sheets and the core layer are postulated. For the face sheets: !

vt2 ¼ ua2 þud2  x3 þ

tc þ tf ðua3;2 þ ud3;2 Þ 2

ð1aÞ

Fig. 1. Geometry of a sandwich beam.

vb2

¼ ua2 ud2 

vt3 ¼ ua3 þ ud3 ;

! t c þt f x3  ðua3;2 ud3;2 Þ 2 vb3 ¼ ua3 ud3

ð1bÞ ð1cÞ

where u2 and u3 denote displacements in the x2 and the x3 directions of a point on the mid-surface, respectively, whereas v2 and v3 signify displacements of any point of the beam; tc and tf denote, respectively, the thickness of the core and of the face sheet. Superscripts a and d signify, respectively, the average and the half-difference of the top and the bottom face sheets’ midsurface displacements utj and ubj. That is uaj 

1 t ðu þubj Þ; 2 j

1 udj  ðutj ubj Þ; j ¼ 2; 3 2

ð2aÞ

and subscripts t and b denote the top and the bottom face sheets, respectively. Furthermore, u3;2  ∂u3 =∂x2 . For the core layer: " # tf 2x3 tf 4ðx3 Þ2 vc2 ¼ ua2  ud3;2  c ud2 þ c x3 ua3;2 þ 1 Ωc2 ð3aÞ 2 t t ðt c Þ2 2x3 vc3 ¼ ua3  c ud3 t

ð3bÞ

here, displacement function Ω2 describes the warping of the core. We note that Eqs. (2) and (3) involve five basic unknown functions, c ua2 ; ud2 ; ua3 ; ud3 ; Ω2 . The displacement field at points on the interfaces between face sheets and the core is assumed continuous. To approximate the geometrical nonlinear effect, the beam's deformation is described in terms of the nonlinear Green–Lagrange strain tensor. The components of the strain tensor are given by c

1 2

ε22 ¼ v2;2 þ ðv3;2 Þ2 ;

1 2

ε33 ¼ v3;3 þ ðv3;3 Þ2 ; γ 23 ¼ v2;3 þ v3;2 þ v3;2 v3;3 ð4Þ

The sandwich beam theory presented above is not restricted to any kind of specific material model. Nevertheless, orthotropic materials are assumed for both face sheet and core layer. The core is further assumed to be weak, i.e., sc22 is negligible [3]. To the 2-D problem, we are addressing the stress–strain relations for the face sheets and the core can be represented as 8 9 2 9 38 ε = Q 22 Q 23 0 > > < s22 > = < 22 > 0 7 s33 ¼ 6 ð5Þ 4 Q 23 Q 33 5 ε33 > > > :s > ; 0 0 Q 44 : γ 23 ; 23 Introducing the concept of reduced stiffness (see e.g.,[11]), the stress–strain relations for the face sheets and the core can be summarized as f

sf22 ¼ Q 22 εf22 ;

sf23 ¼ Q f44 γ f23

sc33 ¼ Q c33 εc33 ;

sc23 ¼ Q c44 γ c23

where the reduced stiffness

ð6aÞ ð6bÞ f Q 22

 Q f22 ½ðQ f23 Þ2 =Q f33 .

244

Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

  2 Nc33 ¼ Q c33 ud3 2 þ c ud3 t

3. Formulation of the governing system The governing equations and the compatible boundary conditions for the beam can be derived by using the extended Hamilton's principle [8]: Z t2 ðδT δU þ δW e Þ dt ¼ 0 ð7aÞ t1

with

δuaj ¼ δudj ¼ δΦcj ¼ δΩc2 ¼ 0 at t ¼ t 1 and t 2

ð7bÞ

where j¼2,3, δT and δU denote the virtual kinetic and the virtual strain energies, respectively, while δW e denotes the virtual work done by external forces. We note here that δW e does not necessarily represent the variation of the work function We, thus underlying the term “extended” in the extended Hamilton's principle as expounded by Meirovitch [8, p. 85]. For the present problem, expressions for δT , δU and δW e are given below. Virtual kinetic energy: δT ¼

Z

(Z

ℓ

t c =2 þ t f t =2 c

0

ρf ðv_ b3 δv_ b3 Þ dx3 þ

Z

t c =2 t =2 c

ρc v_ c3 δv_ c3 dx3 þ

Z

t =2t c

f

f

in which, the membrane stiffness Af22  Q 22 t f , and the bending f Q 22 ðt f Þ3 =12.

stiffness Df22  By invoking the routine process of variational operations, we derive both the governing equations and the compatible boundary conditions, among which the governing equations can be summarized as

δua2 : N a22;2 ¼ 0 δud2 : N d22;2  c N c23 ¼ 0

1 t

ð13bÞ

δΩc2 : Mc23 ¼ 0

ð13cÞ

δua3 : ð2mf0 þmc0 Þu€ a3 þ 2Ma22;22 þ2ðN d22 ud3;2 Þ;2 þ 1þ

ρf ðv_ t3 δv_ t3 Þ dx3 dx2

Virtual strain energy: Z

ℓ

(Z

t c =2

0

Z þ

t c =2 þ t f

t c =2 t c =2

sb22 δεb22 dx3 þ

Z

t c =2

t c =2t f

st22 δεt22 dx3

 mc2 d u€ 3 þ2M d22;22 þ 2ðNd22 ua3;2 Þ;2 þ 2ðN a22 ud3;2 Þ;2 3     2 2 2 þ c N c33 1 c ud3 þN c23 ua3;2 c þ 2qd3 ¼ 0 ð13eÞ t t t

qa3 ¼ ð9Þ

dx2

Here, ρf and ρc are mass densities of the face sheets and the core, respectively. It is noted that the virtual kinetic energy density ρc v_ c2 δv_ c2 and the virtual strain energy density sc22 δεc22 are disregarded due to the adoption of weak core, i.e., only the transverse inertia effects are kept, as discussed by Hohe and Librescu [7]. Virtual work due to external forces: Z ℓ δW e ¼ ðqt3 δut3 þ qb3 δub3 Þ dx2 ð10Þ 0

qt3

qb3

Here, and denote the applied load on the upper and bottom face sheets, respectively. Define the following stress resultants and stress couples: Z t c =2 Z t c =2 þ tf ðN b22 ; M b22 Þ  ð1; x3 Þsb22 dx3 ; ðN t22 ; M t22 Þ  ð1; x3 Þst22 dx3 t c =2t f

t c =2

ð11aÞ N c23 

Z

t c =2

t c =2

ð1; x3 Þsc23 dx3 ;

N c33 

Z

t c =2 t c =2

ð1; x3 Þsc33 dx3

we obtain the following relations:   N t þ Nd22 1 1 ¼ Af22 ua3;2 þ ðua3;2 Þ2 þ ðud3;2 Þ2 N a22  22 2 2 2 N d22

N t N d22 ¼ Af22 ½ud2;2 þ ua3;2 ud3;2   22 2

ð11bÞ

ð12aÞ

ð12bÞ

M a22 

M t22 þ M b22 ¼ Df22 ua3;22 ; 2

ð12cÞ

M d22 

M t22 M b22 ¼ Df22 ud3;22 2

ð12dÞ

N c23 ¼ Q c44 ½2ud2 þ t f ua3;2 þ ua3;2 ðt c 2ud3 Þ;

ð13dÞ

In Eqs. (13a–e) mf0  ρf t f and mc0  ρc t c are the inertial coefficients, whereas qa3 and qd3 , respectively, are defined as

)

ðsc33 δεc33 þ sc23 δγ c23 Þ dx3

!   tf 2 d 2 d c c  u u þ 2qa3 ¼ 0 c c 3 N 23;2 N 23 t t c 3;2 t

δud3 :  2mf0 þ mc0 þ

ð8Þ

δU ¼

ð13aÞ



)

t c =2

ð12fÞ

ð12eÞ

qt3 þ qb3 ; 2

qd3 ¼

qt3 qb3 2

ð14Þ

The corresponding boundary conditions at x2 ¼ 0; ℓ are

δua2 : N a22 ¼ 0 or ua2 ¼ u^ a2

ð15aÞ

δud2 : N d22 ¼ 0 or ud2 ¼ u^ d2

ð15bÞ tf t

2 t

!

δua3 : 2Ma22;2 þ 2N a22 ua3;2 þ 2N d22 ud3;2 þ 1 þ c  c ud3 Nc23 ¼ 0

or

a

ua3 ¼ u^ 3

ð15cÞ

δud3 : Md22;2 þ N d22 ua3;2 þ Na22 ud3;2 ¼ 0 or ud3 ¼ u^ d3

ð15dÞ

δua3;2 : Ma22 ¼ 0 or ua3;2 ¼ u^ a3;2

ð15eÞ

δud3;2 : Md22 ¼ 0 or ud3;2 ¼ u^ d3;2

ð15fÞ

in which, quantities with a superimposed hat are specified at the boundaries. Further simplifications can be performed on Eqs. (13a–e). Eq. (13a) states that N a22 ¼ C is constant along x2 during vibration. In case that both the ends of the beam are immovable in x2 direction, the constant C can be determined as follows. After integrating Eq. (12a) over the beam span, we get Z ℓ Af ½ðua3;2 Þ2 þ ðud3;2 Þ2  dx2 ð16Þ C ¼ 22 2ℓ 0 On the other hand, when either end of the beam is freely movable in x2 direction, we have C¼ 0. For other cases of boundary conditions, see [12] for the determination of the constant C. In the sequel, we will focus on the case that C ¼ 0. It has to be noted that no damping appears in the governing equations (13a–e) and the boundary conditions (15a–f). However, damping is actually present in all realistic structures due to energy dissipation. Furthermore, the nonlinear interactions we are investigating here involve delicate dynamic balance among nonlinearities, damping and external loads. The structural nonlinearities of the sandwich beam have been clearly defined in Eq. (4), and as to

Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

damping, we restrict ourselves to small viscous damping and it is eventually manifested as viscous damping factors in the framework of the classical modal analysis (see e.g., [8, pp. 198–203]). These viscous damping factors can be experimentally determined. In the next section, we will add simplified damping terms after we have discretized the undamped natural system. As pointed out by Chen and Babcock [13], the correct choice of delicate orders of nonlinearities, damping and external loads cannot be made from mathematical reasoning alone but demands prerequisite knowledge of the nature of the solution to be sought. In our case, we will focus on investigating modal interactions via the powerful multiple scales method.

In Eqs. (19a, b), ηα and ηd are the damping coefficients, while B r (r ¼ 15) and C r ðr ¼ 17Þ are defined as Z 1 B 1 ði; mÞ  B⋆ ð20aÞ 1 ðm; X 2 Þ sin ðλi X 2 Þ dX 2 0

Z

1

B 2 ði; mÞ 

B⋆ 2 ðm; X 2 Þ sin ðλi X 2 Þ dX 2

0

Z

1

B 3 ðm; n; iÞ 

B⋆ 3 ðm; n; X 2 Þ sin ðλi X 2 Þ dX 2

0

Z

1

B 4 ðm; n; p; iÞ  0

Z

1

B 5 ðiÞ 

4. Analytical solution via the multiscale method

U d2 

ud2 ua ud qa ; U a3  c 3 f ; U d3  c 3 f ; Q a3  3 ; ℓ b1 t þ 2t t þ 2t

x2 X2  ; ℓ

X3 

x3 t þ 2t c

mf0 , 0c

τ  ωt;

; f

Q d3 

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u Df ω  t f 22c ðm0 þ m0 =2Þℓ4

ð17aÞ

ð17bÞ

m¼1

0

0

m¼1

where λm  mπ , with m being the number of modal waves of either the global bending or face wrinkling. Wma and Wmd denote the generalized coordinates of the corresponding shape function sin ðλm X 2 Þ. Then the solution for the in-plane displacements ua2 and ud2 can be obtained by substituting the representations (18) into Eqs. (13a, b). Once the solutions of ua2 and ud2 have been obtained, the Galerkin weighted residual method (see e.g., [8, pp. 544–547]) is used to discretize the nonlinear partial differential equations (13c, d). Two sets of nonlinearly coupled ordinary differential equations governing the time evolution of Wma and Wmd are represented in the following form: a

m¼1n¼1

( þ

M

∑

N

)

N

∑ B 4 ðm; n; q; iÞW am W dn W dq fB 5 ðiÞg ¼ 0

∑

ð19aÞ

m¼1n¼1q¼1

d

d € g þ η fW _ g þ ½C 2 ði; jÞ ½C 1 ði; jÞNN fW NN fW j g d j j ( ) ( M

þ

∑

d

M

∑ C 3 ðm; p; iÞW am W ap þ

m¼1p¼1

( þ

M

∑

N

M

∑

∑

N

N

m¼1n¼1p¼1

( þ

N

∑

N

∑

N

∑ C 4 ðn; q; iÞW dn W dq

n¼1q¼1

) C 5 ðm; n; p; iÞW am W dn W ap )

∑ ∑ C 6 ðn; q; r; iÞW dn W dq W dr fC 7 ðiÞg ¼ 0

n¼1q¼1r ¼1

)

ð19bÞ

1

C⋆ 3 ðm; p; X 2 Þ sin ðλj X 2 Þ dX 2

C⋆ 4 ðn; q; X 2 Þ sin ðλj X 2 Þ dX 2

Z

1

C 5 ðm; n; p; jÞ  0

Z C 6 ðn; q; r; jÞ 

1

0 1 0

C⋆ 5 ðm; n; p; X 2 Þ sin ðλj X 2 Þ dX 2

C⋆ 6 ðn; q; r; X 2 Þ sin ðλj X 2 Þ dX 2

c1 Q d3 sin ðλj X 2 Þ dX 2

ð20hÞ

ð20iÞ

ð20jÞ

ð20kÞ

ð20lÞ

The associated coefficients B⋆ 1 ; …; c1 are defined in the Appendix. In order to solve the weak nonlinear system (19a, b), the method of multiple scales [14] is used to investigate the nonlinear response. We assume the following asymptotic expansions: W am ¼ εW am1 þ ε2 W am2 þ ε3 W am3 þ ⋯; W dm ¼ εW dm1 þ ε2 W dm2 þ ε3 W dm3 þ ⋯

ð21Þ

where ε is a small dimensionless parameter. We further introduce the following time scales Ti ði ¼ 0; 1; 2; …Þ: T 0  τ;

a € g þ η fW _ g þ ½B 2 ði; jÞ ½B 1 ði; jÞMM fW MM fW j g α j j  M  N þ ∑ ∑ B 3 ðm; n; iÞW am W dn

1

0

Z

ð20eÞ

ð20gÞ

C 4 ðn; q; jÞ 

C 7 ðjÞ 

ð20dÞ

C⋆ 2 ðn; X 2 Þ sin ðλj X 2 Þ dX 2

0

M

U d3 ðX 2 ; τ Þ ¼ ∑ W dm ðτÞ sin ðλm X 2 Þ

ð20cÞ

ð20fÞ

C 3 ðm; p; jÞ 

ð18Þ

a

1

C 2 ðj; nÞ 

ð20bÞ

C⋆ 1 ðn; X 2 Þ sin ðλj X 2 Þ dX 2

Z

in which, and are defined in the preceding section, while b1 is defined in the Appendix. In the sequel, we consider a simply supported sandwich beam. As a result, the transverse displacements can be expanded as M

1

Z

Df22

U a3 ðX 2 ; τÞ ¼ ∑ W am ðτÞ sin ðλm X 2 Þ;

Z C 1 ðj; nÞ  Z

qd3 b1

B⋆ 4 ðm; n; p; X 2 Þ sin ðλi X 2 Þ dX 2

b1 Q a3 sin ðλi X 2 Þ dX 2

0

In order to solve the governing equations (13a–e) together with the boundary conditions (15a–f), we at first define the following basic dimensionless parameters:

245

T 1  ετ;

T 2  ε2 τ ; …

ð22Þ

For the problem we are investigating, we assume that ηα ¼ εη⋆ 1, ηd ¼ εη⋆2 , and the external loads Q a3 and Q d3 are on the scale of Oðϵ2 Þ. Following the routine process of the multiscale method, we obtain, for each power of ε, two sets of equations as follows: On scale OðϵÞ: ( ) ∂2 W aj1 ð23aÞ ½B 1 ði; jÞ þ ½B 2 ði; jÞMM fW aj1 g ¼ 0 ∂T 20 ( ) ∂2 W dj1 þ ½C 2 ði; jÞNN fW dj1 g ¼ 0 ½C 1 ði; jÞ ∂T 20 On scale Oðϵ2 Þ: ( ) ( ) ∂2 W aj1 ∂2 W aj2 ∂W aj1 ⋆ þ η ½B 1 ði; jÞ 2 þ þ ½B 2 ði; jÞfW aj2 g 1 ∂T 0 ∂T 1 ∂T 0 ∂T 20  M  N þ ∑ ∑ B 3 ðm; n; iÞW am1 W dn1 fB 5 ðiÞg ¼ 0 m¼1n¼1

ð23bÞ

ð24aÞ

246

Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

(

∂2 W dj1

∂2 W dj2

) þη

½C 1 ði; jÞ 2 þ ∂T 0 ∂T 1 ∂T 20 ( þ ½C 2 ði; jÞfW dj2 g þ ( þ

N

N

∑

∑

n¼1q¼1

( ) ∂W dj1

⋆ 2

M

M

∑

∑

m¼1p¼1

∂T 0

here, s1 is another detuning parameter which is introduced to quantify the nearness of the excitation frequency to the primary resonance frequency ωak . Substituting expressions in Eq. (27) into Eqs. (24a, b), the absence of secular terms leads to the following solvability conditions:

)

C 3 ðm; p; iÞW am1 W ap1 )

C 4 ðn; q; iÞW dn1 W dq1

fC 7 ðiÞg ¼ 0

ð24bÞ g 7 eis1 T 1 þig 1

On scale Oðϵ Þ: ( ) ∂2 W aj1 ∂2 W aj1 ∂2 W aj2 ∂2 W aj3 ½B 1 ði; jÞ 2 þ þ 2 þ þ ½B 2 ði; jÞfW aj3 g ∂T 0 ∂T 2 ∂T 0 ∂T 1 ∂T 21 ∂T 20 ( )   M N ∂W aj2 ∂W aj1 þ η⋆ þ þ ∑ ∑ B 3 ðm; n; iÞðW am1 W dn2 þ W am2 W dn1 Þ 1 ∂T 0 ∂T 1 m¼1n¼1 3

(

M

þ

∑

N

)

N

∑ B 4 ðm; n; q; iÞW am1 W dn1 W dq1

∑

¼0

ð25aÞ

m¼1n¼1q¼1

( ½C 1 ði; jÞ 2 ( þ η⋆ 2

( þ

∂2 W dj1 ∂T 0 ∂T 2

∂W dj2 ∂T 0

N

∑

þ

þ

∂2 W dj1 ∂T 21

(

) d

∂W j1 ∂T 1

þ2

þ

M

∂2 W dj2 ∂T 0 ∂T 1

∑

þ

∂2 W dj3

þ

M

∑

∑

( þ

N

∑

N

m¼1p¼1

)

∑ ∑

n¼1q¼1r ¼1

) ¼0

ð25bÞ

0

1 0

Z

ω

d n T 0 Þ þ c:c:

ð26Þ

where c.c. represents the complex conjugate of the preceding terms, ωam and ωdn are the natural frequencies obtained by the linear modal analysis. At this stage, Z am and Z dn are unknown, and they are to be determined at the next level of approximation by fulfilling the solvability conditions (see e.g., [14, pp. 388–462]). Substituting the solution form (26a, b) into Eqs. (24a, b), we can obtain several internal resonance combinations which may lead to secular terms with respect to the quadratic nonlinearity, for example, ωak  ωaα þ ωdβ and ωdk  ωdα þ ωdβ . On the other hand, if no internal resonance combination occurs in the second scale, we can go to the third scale to search for the internal resonance. With respect to cubic nonlinearity, we can find the internal resonance combinations, such as ωak  ωaα 7 ωaβ 7 ωaγ , ωak  ωaα 7 ωdβ 7 ωdγ , and ωdk  ωdα 7 ωdβ 7 ωdγ . Apparently, 1:1, 2:1, and 3:1 internal resonances are the special cases of the preceding combinations when β ¼ γ or α ¼ β ¼ γ . For further investigation of modal interactions in the presence of internal resonance, we will focus on the case of combination ωak  ωaα þ ωdβ . Detuning parameter s2 is introduced such that εs2 ¼ ωak ðωaα þ ωdβ Þ. 4.1. Case I: excitation energy input from the mode of ωak In such a case, the external load of the sandwich beam is assumed to be the primary resonance in the form: Q a3 ¼ F expðiΩT 0 Þ þ c:c:; Q d3 ¼ 0; Ω ¼ ωak þ εs1 ;

Z g6 

W am1 ¼ Z am ðT 1 ; T 2 ; ⋯Þexpðiωam T 0 Þ þ c:c:; ¼ Z dn ðT 1 ; T 2 ; ⋯Þexpði

1

g 5  2ωdβ

The solution of Eqs. (23a, b) can be written in the form:

W dn1

ð28cÞ

with F ¼ ε2 f 2 ð27Þ

B⋆ 3 ð; α; β ; X 2 Þ sin ðkπ X 2 Þ dX 2

g 3  2ωaα Z

)

C 6 ðn; q; r; iÞW dn1 W dq1 W dr1

1 0

g4 

∑ C 5 ðm; n; q; iÞW am1 W dn1 W ap1

M

a

in which, the coefficients gj ðj ¼ 1; 7Þ are defined as Z 1 B⋆ g 1  2ωak 1 ðk; X 2 Þ sin ðkπ X 2 Þ dX 2 ;

)

M

m¼1n¼1p¼1

ð28bÞ

d d þ g 6 Z ak Z α eis2 T 1 iη⋆ 2 ωβ Z β ¼ 0

Z

∑ C 3 ðm; p; iÞðW am1 W ap2 þ W am2 W ap1 Þ

M

∂T 1

g2 

∑ C 4 ðn; q; iÞðW dn1 W dq2 þ W dn2 W dq1 Þ N

∂Z dβ

ð28aÞ

ð28aÞ

0

þ½C 2 ði; jÞfW dj3 g

n¼1q¼1

(

ig 5

∂Z aα d a a þ g 4 Z ak Z β eis2 T 1 iη⋆ 1 ωα Z α ¼ 0 ∂T 1

)

∂T 20

N

ig 3

∂Z ak a a þ g 2 Z aα Z dβ eis2 T 1 iη⋆ 1 ωk Z k ¼ 0 ∂T 1

g7 

1 0

Z

1

0

ð28bÞ

B⋆ 1 ðα; X 2 Þ sin ðαπ X 2 Þ dX 2 ;

ð28cÞ

B⋆ 3 ðk; β ; X 2 Þ sin ðαπ X 2 Þ dX 2 Z

1 0

ð28dÞ

B⋆ 1 ðβ ; X 2 Þ sin ðβπ X 2 Þ dX 2 ;

ð28eÞ

C⋆ 3 ðk; α; X 2 Þ sin ðβπ X 2 Þ dX 2

ð28fÞ

b1 f 2 sin ðkπ X 2 Þ dX 2

ð28gÞ

It is convenient to express the unknown quantities in Eqs. (28a–c) in polar form: Z ak ¼ α1 ðT 1 Þeiβ1 ðT 1 Þ ;

Z aα ¼ α2 ðT 1 Þeiβ2 ðT 1 Þ ;

Z dβ ¼ α3 ðT 1 Þeiβ3 ðT 1 Þ

ð29Þ

Substituting expressions in Eq. (29) to Eqs. (28a–c) and separating the real parts from the imaginary parts, we get

α_ 1 ¼ ðg 2 α2 α3 sin γ η⋆1 ωak α1 g7 sin ξÞ=g 1

ð30aÞ

α1 β_ 1 ¼ ðg2 α2 α3 cos γ g7 cos ξÞ=g1

ð30bÞ

α_ 2 ¼ ðg 4 α1 α3 sin γ η⋆1 ωaα α2 Þ=g 3

ð30cÞ

α2 β_ 2 ¼ g 4 α1 α3 cos γ =g3

ð30dÞ

α_ 3 ¼ ðg 6 α1 α2 sin γ η⋆2 ωdβ α3 Þ=g5

ð30eÞ

α3 β_ 3 ¼ g 6 α1 α2 cos γ =g5

ð30fÞ

in which, γ ¼ β 2 þ β3 þ s2 T 1 β1 , ξ ¼ β1 s1 T 1 . In the sequel, we a ⋆ a ⋆ d further define ζ 1  η⋆ 1 ωk =g 1 , ζ 2  η1 ωα =g 3 , and ζ 3  η2 ωβ =g 5 . The steady-state motions occur when α_ j ¼ 0 (j ¼ 1; 2; 3), γ_ ¼ 0, and ξ_ ¼ 0, which correspond to the fixed points solution of Eqs. (30a–f). There are three cases of the fixed points solution in Eqs. (30a–f): (a) Trivial solution α1 ¼ α2 ¼ α3 ¼ 0. (b) Nontrivial solution α2 ¼ α3 ¼ 0, α1 a 0. In such a case, α1 and ξ are governed by the following

Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

247

Fig. 2. Finite element mesh used in the modal analysis by the ABAQUS. It consists of 400  2 elements in each of the top and bottom face sheets, and 400  18 elements in the core.

Table 2 Accuracy test of the first 10 Eigenfrequencies ωam [Hz] of the sandwich beam. m

1

2

3

4

5

144.99 143.02 1.4 8

229.27 227.93 0.6 9

336.57 335.41 0.3 10

999.64 998.01 0.2

1224.7 1222.0 0.2

4

5

Theoretical result By ABAQUS Relative errora (%) m

34.54 34.06 1.4 6

81.28 79.42 2.3 7

Theoretical result By ABAQUS Relative error (%)

465.83 466.0 0.04

621.10 619.93 0.2

a

799.72 797.28 0.3

Relative error, ([prediction by the present model]  [result by ABAQUS])/[result by ABAQUS]100%.

Table 3 Accuracy test of the first 10 Eigenfrequencies ωdn (Hz) of the sandwich beam. m

1

2

3

Theoretical result By ABAQUS Relative errora (%) m

753.81 752.58 0.2 6

755.28 753.07 0.3 7

761.60 755.89 0.8 8

778.37 768.94 1.2 9

812.58 799.25 1.7 10

Theoretical result By ABAQUS Relative error (%)

871.37 854.47 2.0

960.37 940.58 2.1

1082.8 1060.8 2.0

1239.4 1215.6 2.0

1429.5 14040.0 1.8

a

Relative error, ([prediction by the present model]  [result by ABAQUS])/[result by ABAQUS]100%.

Fig. 3. First three and a high-order modes of the sandwich beam computed by the ABAQUS.

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Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

α1 β_ 1 ¼

equations:

ζ 1 α1 þ g7 sin ξ ¼ 0;

ð31aÞ

g7 cos ξ ¼ 0 g1

ð31bÞ

s1 α1 þ

The frequency–response relation can be represented as  2 g7 2 ðζ 1 þs21 Þα21 ¼ g1

ð32Þ

α_ 3 ¼

In such a case, we define the detuning parameter s1 as

ð35cÞ

g4 g α1 α3 cos γ  7 cos ξ g3 g3

ð35dÞ

g6 α1 α2 sin γ ζ 3 α3 g5

α3 β_ 3 ¼

ð35eÞ

g6 α1 α2 cos γ g5

ð35fÞ

in which, ξ ¼ β2 s1 T 1 , γ , ζ 1 , ζ 2 and ζ 3 are defined in Case I. Similarly, there are three cases of the solution of the fixed points in Eqs. (30a–f): (a) Trivial solution α1 ¼ α2 ¼ α3 ¼ 0. (b) Nontrivial solution α1 ¼ α3 ¼ 0, α2 a 0. response relation can be represented as  2 g7 2 ðζ 2 þ s21 Þα22 ¼ g3

4.2. Case II: excitation energy input from the mode of ωaα

ð35bÞ

g4 g α1 α3 sin γ ζ 2 α2  7 sin γ g3 g3

α2 β_ 2 ¼

It is interesting to note that the fixed point solution in this case degenerates to a linear solution. (c) Nontrivial solution α1 a0, α2 a0, α3 a 0. After some manipulations, the frequency–response relation in this case can be represented as   g4 g6 s s 2 α21  1 2 ¼ 1 ð33Þ g3 g5 ζ2 ζ3 ζ2 þ ζ3

εs1 ¼ Ωωaα

α_ 2 ¼

g2 α2 α3 cos γ g1

The

frequency–

ð36Þ

ð34Þ

Parallel to Eqs. (30a–f), the solvability conditions in this case can be represented as g g1

α_ 1 ¼  2 α2 α3 sin γ ζ 1 α1

ð35aÞ

Fig. 6. Saturation phenomenon in the response amplitude versus excitation intensity f2 plot ðs1 ¼ s2 ¼ 0Þ. Fig. 4. Steady-state evolution of α1 , α2 and α3 with time when the excitation energy is input from the mode of ωa9 (s1 ¼ s2 ¼ 0, and f 2 ¼ 1).

Fig. 5. Steady-state evolution of α1 , α2 and α3 in the case that the excitation energy is input from the mode of ωa1 (s1 ¼ s2 ¼ 0, and f 2 ¼ 1).

Fig. 7. Jump phenomenon in the response amplitude versus frequency s1 plot (f 2 ¼ 1, s2 ¼ 0).

Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

(c) Nontrivial solution α1 a0, α2 a 0, α3 a 0. The frequency– response relation can be represented as   g2 g6 s þs 2  α22  1 2 ¼ 1 ð37Þ g1 g5 ζ 1 ζ3 ζ1 þ ζ3 Standard eigenvalue analysis of the Jacobian matrix at each of the fixed points is conducted to determine the stability of the corresponding fixed points (see e.g., [8, pp. 167–169]).

5. Results and discussion For a concrete study of modal interaction caused by the internal resonance, we consider a sandwich beam with length ℓ ¼ 500 mm. The face sheets are taken to be isotropic, with Young's modulus Ef ¼70 GPa, Poisson's ratio νf ¼ 0:3, density ρf ¼ 3700 kg=m3 , and thickness tf ¼1 mm. For the core layer, it is also assumed to be isotropic, with Young's modulus of Ec ¼0.7 MPa, Poisson's ratio of νc ¼ 0:3, density of ρc ¼ 3:7 kg=m3 , thickness of tc ¼18 mm. Then the natural frequencies of this sandwich beam can be calculated by the eigenvalue analyses of Eqs. (23a, b). In order to test the accuracy of our model, we use the commercial code ABAQUS for modal analysis, which is also used in simulating the forced nonlinear vibration responses of the sandwich beam in the sequel. The discretization of the sandwich beam into finite elements is shown in Fig. 2, in which the top and bottom face sheets are discretized into 400  2 elements, whereas the core is discretized into 400  18 elements. An eight-node biquadratic plane stress element type CPS8R is adopted in the computation. In order to implement the simply supported boundary conditions, u2 ¼ u3 ¼ 0 at the middle nodes of the top and bottom face sheets, and of the core on both the ends are enforced. The external loads are divided equally into two parts and distributed uniformly on both the top and bottom face sheets. Theoretical prediction of the natural frequencies and the results from ABAQUS are compared in Tables 2 and 3. In all computed cases, an excellent agreement is observed

249

between the theoretical and numerical results. Fig. 3 further displays the first three and a high-order mode shapes of ua3 and ud3 by ABAQUS. Next, we choose a nearly commensurable eigenfrequency combination ωa1 þ ωd7  ωa9 (i.e., α ¼ 1; β ¼ 7, k¼9) for the investigation. In such a case, the coefficients in Eqs. (30a–f) are specified as g1 ¼ 3808.26, g2 ¼60189.15, g3 ¼ 131.08, g4 ¼60189.15, g5 ¼ 3457.09, g6 ¼ 735.21, g7 ¼0.32. The damping coefficients (dimensionless) are specified as ζ 1 ¼ ζ 2 ¼ ζ 3 ¼ 0:01. The 4th-order Runge–Kutta method is used to solve Eqs. (30a–f). When the excitation energy is directly input from the mode of ωa9 (referred to as the α1 mode, see Eq. (29)), the responses of α1 , α2 and α3 are shown in Fig. 4, in which we see that the modes of ωa1 and ωd7 are also excited, and in steady-state, all the three modes are also present. This is consistent with the third theoretical solution of Case I in the preceding analysis. Similarly, when the excitation energy is directly input from the mode of ωa1 (referred to as the α2 mode, see Eqs. (29)), the responses of α1 , α2 and α3 are shown in Fig. 5. Quite different to the result in Fig. 4, almost no modal interaction exists in this case and the steady-state responses of α1 and α3 disappear in Fig. 5. This is precisely the second theoretical solution α2 a0, α1 ¼ α3 ¼ 0 in Case II. Now, one intriguing question arises: how is it that no modal interaction exists in Fig. 5 while strong modal interactions are present in Fig. 4, since the involved nonlinear dynamic system retain the same except with different excitations? To answer this question, we turn to Eq. (37). With the parameters gm and ζ n specified at the beginning of this section, it turns out that g 2 g 6 =ðg 1 g 5 ζ 1 ζ 3 Þ 4 0. Therefore, Eq. (37) is not fulfilled and the third theoretical solution α2 a 0, α1 ¼ α3 ¼ 0 in Case II is physically infeasible. In the sequel, we will specifically investigate whether or not the nonlinear dynamic system (30) can only have the third theoretical solution (Case I) when excitation energy is input from the α1 mode. For this purpose, we conduct parametric analysis of the forced responses with respect to f2 and s1 . Fig. 6 shows the steady-state

Fig. 8. The forced response of the sandwich beam when the excitation energy is input from the mode of ωa9 , (a) response of ut3 ðℓ=2; tÞ, (b) response of ub3 ðℓ=2; tÞ, (c) FFT of ua3 ðℓ=2; tÞ, (d) FFT of ud3 ðℓ=2; tÞ.

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Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

evolution of αj versus f2 with s1 ¼ s2 ¼ 0. It is readily seen that when f 2 o 0:12, the nonlinear dynamic system (30) has the second theoretical solution α1 a 0, α2 ¼ α3 ¼ 0; when f 2 4 0:12, the system has the third theoretical solution α1 a 0, α2 a 0, α3 a 0. Interestingly, when f 2 4 0:12, α1 retains a specific amplitude and does not vary with respect to f2. We call that the α1 mode is subject to saturation. Keeping f 2 ¼ 1 and s2 ¼ 0, while varying s1 from  0.1 to 0.1, we get the steady-state evolution of αj versus s1 in Fig. 7. It is seen that s1 increases from 0, the system (see Eqs. (30a–f)) has the third theoretical solution α1 a 0, α2 ¼ α3 ¼ 0. When s1 reaches 0.038, the system does not transit to the second theoretical solution but keep the third theoretical solution. However, when the detuning parameter s1 increases beyond 0.056, α1 jumps to its second theoretical solution, while α2 and α3 jump to zero. Finally, the commercial code ABAQUS is used to validate our theoretical results on modal interactions of the sandwich beam. The excitation in Case I (see Eq. (27)) is at first used in simulation of undamped forced vibration response of the sandwich beam by ABAQUS, in which the uniformly distributed load is specified as qa3 ¼ 0:015 N=mm2 , qd3 ¼ 0, and Ω ¼ 998:01 Hz. The responses at the mid-span of the mid-surface of the top and bottom face sheets (i.e., ut3 ðℓ=2; tÞ and ub3 ðℓ=2; tÞ) are obtained and displayed in Fig. 8a and b. From Eq. (2), ua3 ðℓ=2; tÞ and ud3 ðℓ=2; tÞ can be determined, then fast Fourier Transform (FFT) is used for the frequency-domain analysis, results of which are displayed in Fig. 8c and d. In Fig. 8c, the presence of spectrum peak A1 is due to the modal interactions and it is the α2 mode, with frequency 34.06 Hz; spectrum peak B1 is the resonant α1 mode with frequency 998.01 Hz. In Fig. 8d, the presence of spectrum peak D1 is due to the modal interactions and it is the α3 mode, with frequency 940.58 Hz, however, the presence of spectrum peak C1 is not due to modal interactions of the combination internal resonance ωa9  ωa1 þ ωd7 . High ratio of the amplitude at A1 over the amplitude at B1 signifies one remarkable feature of modal interaction of the sandwich beam, i.e., low-amplitude high-frequency excitation can induce highamplitude low-frequency response, which is very dangerous in

practice. The simulation results are consistent with the third theoretical solution α1 a 0, α2 a 0, α3 a 0 of Case I. It is interesting to note that the interaction between face sheets’ high-frequency oscillations and the sandwich beam's global low-frequency oscillations can lead to very complex vibration behavior, and even chaotic, unpredictable response, as shown by Hohe et. al. [15] The excitation in Case II (34) is then used in the simulation of undamped forced vibration response of the sandwich beam. The uniformly distributed load is specified as qa3 ¼ 0:015 N=mm2 , qd3 ¼ 0, and Ω ¼ 34:06 Hz. The responses at the mid-span of the mid-surface of the top and bottom face sheets (i.e., ut3 ðℓ=2; tÞ and ub3 ðℓ=2; tÞ) are obtained and displayed in Fig. 9a and b. FFT is used for the corresponding frequency-domain analysis, results of which are displayed in Figs. 9c and 7d. In Fig. 9c, the presence of spectrum peak A2 is the resonant α2 mode with frequency 34.06 Hz, but the α1 modes due to modal interactions is not present. In Fig. 9d, the α3 mode does not appear neither. The simulation results are consistent with the second theoretical solution α2 a 0, α1 ¼ α3 ¼ 0 of Case II . These simulations thus validate our theoretical analysis in the preceding section.

6. Conclusions A {2, 1}-order theory is used to derive the governing system of a sandwich beam incorporating the transversely compressibility and geometrical nonlinearity. Via the multi-mode Galerkin method and the multiscale approach, the modal interactions due to combination internal resonance and subjected to primary-resonance excitation are investigated. Commercial code ABAQUS validates our theoretical analysis. Major conclusions include: (i) Multiple internal resonances can simultaneously exist for a sandwich beam with weak and thick core. (ii) The saturation and jump phenomena in the steady-state response of the sandwich beam due to modal interactions are found and investigated via the parametric analyses.

Fig. 9. The forced response of the sandwich beam when the excitation energy is input from the mode of ωa1 , (a) response of ut3 ðℓ=2; tÞ, (b) response of ub3 ðℓ=2; tÞ, (c) FFT of ua3 ðℓ=2; tÞ, (d) FFT of ud3 ðℓ=2; tÞ.

Z. Qin et al. / Thin-Walled Structures 73 (2013) 242–251

(iii) Due to interactions between global and local modes of a sandwich beam with weak and thick cores, low-amplitude high-frequency excitation can induce high-amplitude lowfrequency response and can lead to disastrous consequence.

251

B⋆ 1 ðm; X 2 Þ  b1 sin ðλm X 2 Þ ⋆ 4 2 B⋆ 2 ðm; X 2 Þ  b2 ðλm Þ sin ðλm X 2 Þ þ b3 ðλm Þ sin ðλm X 2 Þ þ b4 A2;2 ⋆ ⋆ 2 B⋆ 3 ðm; n; X 2 Þ  b4 A1;2 þ b5 A2;2 ðλm Þ sin ðλn X 2 Þ

b7 ðλm Þ2 sin ðλm X 2 Þ sin ðλn X 2 Þ þ b8 λm cos ðλm X 2 Þλn cos ðλn X 2 Þb9 A⋆ 2;2 sin ðλn X 2 Þ

Acknowledgment

b12 A⋆ 2 λn cos ðλn X 2 Þ

Partial support from the National Basic Research Program of China with Grant no. 2011CB610305 and the National Natural Science Foundation of China with Grant no. 11021202 is gratefully acknowledged.

⋆ 2 B⋆ 4 ðm; n; q; X 2 Þ  b5 A1;2 ðλq Þ sin ðλq X 2 Þ

þ b6 λm cos ðλm X 2 Þλn cos ðλn X 2 Þðλq Þ2 sin ðλq X 2 Þ 2 b9 A⋆ 1;2 sin ðλq X 2 Þ þb10 ðλm Þ sin ðλm X 2 Þ

 sin ðλn X 2 Þ sin ðλq X 2 Þb11 λm cos ðλm X 2 Þλn  cos ðλn X 2 Þ sin ðλq X 2 Þb12 A⋆ 1 λq cos ðλq X 2 Þ

Appendix. Definition of the coefficients in Section 4 a1  Af22 =ℓ; a3 

a2  2Q c44 ℓ=t c c Q 44 ðt c þ t f Þðt c þ 2t f Þ ; a4 c

t ℓ Af22 ðt c þ 2t f Þ2 ; a5  ℓ3

⋆ ⋆ ⋆ in which, A⋆ 1;2  ∂A1 =∂X 2 , A2;2  ∂A2 =∂X 2 .

Q c44 ðt c þ 2t f Þ2 tc ℓ f c f 2 A ðt þ2t Þ a6  22 ℓ3

b1  ðmf0 þ mc0 =2Þω2 ðt c þ 2t f Þ;

2

C⋆ 1 ðn; X 2 Þ  c1 sin ðλn X 2 Þ 4 C⋆ 2 ðn; X 2 Þ  c2 ðλn Þ sin ðλn X 2 Þ þ c3 sin ðλn X 2 Þ ⋆ 2 C⋆ 3 ðm; p; X 2 Þ  c4 A2;2 ðλp Þ sin ðλp X 2 Þ

Df ðt c þ 2t f Þ b2  22 4 ℓ

C⋆ 4 ðn; q; X 2 Þ  c6 sin ðλn X 2 Þ sin ðλq X 2 Þ

Q c44 ðt c þ t f Þ2 ðt c þ 2t f Þ Q c ðt c þ t f Þ ; b4  44 c c 2 t 2t ℓ Af22 ðt c þ 2t f Þ Af22 ðt c þ 2t f Þ3 b5  ; b6  ℓ2 ℓ4 c c f 2 c Q 44 ðt þ 2t Þ ðt þ t f Þ Q c ðt c þ 2t f Þ2 ðt c þ t f Þ b7  2 ; b8  3 44 c 2 t ℓ t c ℓ2 Q c44 ðt c þ 2t f Þ Q c44 ðt c þ 2t f Þ3 b9  2 ; b10  2 tc t c ℓ2 c c c f 3 Q ðt þ 2t Þ Q ðt c þ2t f Þ b11  6 44 c 2 ; b12  4 44 c t t ℓ b3 

Df22 ðt c þ 2t f Þ ℓ4 Af22 ðt c þ 2t f Þ Q c33 ðt c þ 2t f Þ c3  2 ; c4  tc ℓ2 f c c f 3 A ðt þ 2t Þ Q ðt c þ 2t f Þ2 c5  22 ; c6  6 33 c 2 ℓ4 ðt Þ c1  ðmf0 þ mc0 =6Þω2 ðt c þ 2t f Þ;

c7  4

c2 

Q c33 ðt c þ 2t f Þ3 ðt c Þ3

rffiffiffiffiffi  2 2 a2 a4 λm þ a5 λm λn þ a6 λm λn A1 ðm; n; X 2 Þ  sin ½ðλn þ λm ÞX 2 sinh pffiffiffiffiffiffiffiffiffiffi X2 a1 2 a1 a2 rffiffiffiffiffi  2 2 a2 a4 λm þa5 λm λn a6 λm λn pffiffiffiffiffiffiffiffiffiffi X2 þ sin ½ðλn λm ÞX 2 sinh a1 2 a1 a2 a3 sin ðλm X 2 Þcosh A2 ðm; x2 Þ   2 2 a4 λm þ a5 λm λn þ a6 λm λn rffiffiffiffiffi  a3 A3 ðX 2 Þ  sinh X2 a1 A1 ðm; n; 1ÞA3 ðX 2 Þ A⋆ 1 ðm; n; X 2 Þ  A1 ðm; n; X 2 Þ A3 ð1Þ A2 ðm; 1ÞA3 ðX 2 Þ ⋆ A2 ðm; n; X 2 Þ  A2 ðm; X 2 Þ A3 ð1Þ

rffiffiffiffiffiffiffiffiffiffiffi a2 X2 a1

⋆ 2 C⋆ 5 ðm; n; p; X 2 Þ  c4 A1;2 ðλp Þ sin ðλp X 2 Þ

þ c5 λm cos ðλm X 2 Þλn cos ðλn X 2 Þðλp Þ2 sin ðλp X 2 Þ C⋆ 6 ðn; q; r; X 2 Þ  c7

sin ðλn X 2 Þ sin ðλq X 2 Þ sin ðλr X 2 Þ

References [1] Vinson J. Sandwich structures. Applied Mechanics Reviews 2001;54:201–14. [2] Hohe J, Librescu L. Advances in the structural modeling of elastic sandwich panels. Mechanics of Advanced Materials and Structures 2004;11:395–424. [3] Librescu L. Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Monographs and textbooks on mechanics of solids and fluids: mechanics of elastic stability. Leyden: Noordhoff International; 1975. p. 493–5. [4] Sokolinsky V, Frostig Y. Nonlinear behavior of sandwich panels with a transversely flexible core. AIAA Journal 1999;37(11):1474–82. [5] Nayfeh A. Nonlinear interactions: analytical, computational, and experimental methods, Wiley series in nonlinear science. New York: John Wiley & Sons, Inc.; 2000. [6] Raville M, Ueng C. Determination of natural frequencies of vibration of a sandwich plate. Experimental Mechanics 1967;7:490–3. [7] Hohe J, Librescu L. A nonlinear theory for doubly curved anisotropic sandwich shells with transversely compressible core. International Journal of Solids and Structures 2003;40:1059–88. [8] Meirovitch L. Principles and techniques of vibrations. Upper Saddle River, NJ: Prentice Hall; 1997. [9] ABAQUS, ABAQUSs analysis user's manual, version 6.10. [10] Barut A, Madenci E, Heinrich J, Tessler A. Analysis of thick sandwich construction by a {3,2}-order theory. International Journal of Solids and Structures 2001;38:6063–77. [11] Jones R. Mechanics of composite materials. Washington, DC: Scripta Book Company; 46. [12] Sciuva M, Gherlone M, Librescu L. Implications of damaged interfaces and of other non-classical effects on the load carrying capability of multilayered composite shallow shells. International Journal of Non-Linear Mechanics 2002;37:851–67. [13] Chen J, Babcock C. Nonlinear vibration of cylindrical shells. AIAA Journal 1975;13(7):868–76. [14] Nayfeh A. Introduction to perturbation techniques. John Wiley & Sons Inc.; 1981. [15] Hohe J, Librescu L, Oh S. Dynamic buckling of flat and curved sandwich panels with transversely compressible core. Composites Structure 2006;74:10–24.